Exchange-correlation second derivatives

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

A second major problem connected to the use of finite grids for the evaluation of the exchange-correlation energy is associated with the determination of derivatives of the energy, such as the gradients used in geometry optimizations. We use... 

Note that e disappearance of the exchange correlation derives from the orthogonality of the 0 and p spin functions, which causes the second integral in the second equality to be zero when integrated over either spin coordinate. 

However, in the second set of data, reporting scans of the PES for a limited set of small molecules, it appears that the geometries obtained are satisfactory. Moreover, the nature of the technique used for the determination of Exc, namely the use of a "senior" Exe functional, or the use of the virial theorem, as well as the use of a line integration (not reported here), leads to quite similar geometries. This point is in accord with a similar conclusion obtained by van Gisbergen et ol. in their frequency-dependent polarizabilities [75] they choose to use a "mixed scheme" where a different approximation for fxc and Vxc were used, whereas fxc is the functional derivative of the exchange-correlation potential Vxc, with respect to the time-dependent density. 

Following the work of Garza and Robles [39], the kinetic hardness term is zero and the local hardness depends only on the second derivative of the Coulomb and exchange-correlation energies. The exchange-correlation contribution to T ij is much... 

The analytic form of the first two terms in the Kohn-Sham effective potential (Vrff [p](r)) is known. They represent the external potential (vext which is the nuclear attraction potential in most cases) and Coulomb repulsion between electrons. The second term is an explicit functional of electron density. The last term, however, represents the quantum many-body effects and has a traditional name of exchange-correlation potential. vxc is the functional derivative of the component of the total energy functional called conventionally exchange-correlation energy (Exc[p]) ... 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

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